The away side were ahead inside 20 minutes when Alex Mowatt's cross was met by Matt Smith, who headed home past keeper Ross Turnbull.
Smith almost doubled the visitors' lead moments before the break when he rose highest in the area, but could not keep his header down.
Doncaster went close to an equaliser midway through the second half when Woods's shot from outside the box flew narrowly wide.
Chris Brown tried to find a goal for Rovers moments later, but his effort from the edge of the area was wayward.
Leeds did find a second with 14 minutes remaining thanks to Ross McCormack's 11th goal in seven games, with his close-range shot too good for Turnbull.
With three minutes left, Rodolph Austin netted to guarantee three points for the Elland Road outfit.